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Hermite’s constant for quadratic number fields. (English) Zbl 1042.11045

For the Klein modular group \(z'=(az+b)/(cz+d)\), the fundamental domain has the “floor” inequality for some \(z'\) equivalent to \(z\), \(\text{Im}(z')= \text{Im} (z)/ | cz+d|^2\geq\sqrt 3/2\). This is actually Gauss’s result on positive definite forms that for some nontrivial integer pair \((c,d)\), \(rc^2+scd+td^2 \leq \sqrt{(4rt-s^2)}/ \sqrt 3\). The critical point \(z'=(1+\sqrt{-3})/2\) lies at the intersection of certain component isometric loci \(| cz+d|=1\).
Extensions to the Hilbert modular group involve variables \(z_i\) \((i=1,\dots,n)\) transformed unimodularly over integers of the real conjugate fields \(k_i\) \((i-1,\dots, n)\), then the “floor” inequality would involve the product \(\text{Im} (z_1)\dots\text{Im} (z_n)\). This calls for a generalization of Gauss’s theorem to the product of forms over integers in \(k_i\), formalized by P. Humbert [Comment. Math. Helv. 12, 263–306 (1940; Zbl 0023.19905)]. The reviewer proved that the isometric surfaces can be reduced to a finite set [Math. Comput. 19, 594–605 (1965; Zbl 0144.28501)] (which the authors relate to a finite set of extreme Humbert forms). By a sophisticated reduction process they verify conjectures on the floor components by the reviewer for \(k=\mathbb{Q}(\sqrt m)\), \(m=2,3,5\).

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11R11 Quadratic extensions
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References:

[1] DOI: 10.1090/S0002-9939-97-03940-3 · Zbl 0896.11014 · doi:10.1090/S0002-9939-97-03940-3
[2] DOI: 10.1090/S0025-5718-1965-0195818-4 · doi:10.1090/S0025-5718-1965-0195818-4
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[5] DOI: 10.1007/BF01448854 · doi:10.1007/BF01448854
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